IOP PUBLISHING PLASMA PHYSICS AND CONTROLLED FUSION

Plasma Phys. Control. Fusion 52 (2010) 025002 (17pp) doi:10.1088/0741-3335/52/2/025002

On the requirements to control neoclassical tearingmodes in burning plasmas

O Sauter1,4, M A Henderson1,5, G Ramponi2, H Zohm3 and C Zucca1

1 Ecole Polytechnique Federale de Lausanne (EPFC), Centre de Recherches en Physique desPlasmas, Association EURATOM-Confederation Suisse, Station 13, CH-1015 Lausanne,Switzerland2 Istituto di Fisica del Plasma, EURATOM-ENEA-CNR Association, 20125 Milano, Italy3 IPP-Garching, Max Planck-Institute fur Plasmaphysik, D-85748 Garching, Germany

E-mail: olivier.sauter@epfl.ch

Received 11 September 2009, in final form 16 November 2009Published 18 January 2010Online at stacks.iop.org/PPCF/52/025002

AbstractNeoclassical tearing modes (NTMs) are magnetic islands which increase locallythe radial transport and therefore degrade the plasma performance. Theyare self-sustained by the bootstrap current perturbed by the enhanced radialtransport. The confinement degradation is proportional to the island widthand to the position of the resonant surface. The q = 2 NTMs are muchmore detrimental to the confinement than the 3/2 modes due to their largerradii. NTMs are metastable in typical scenarios with βN � 1 and in theregion where the safety factor is increasing with radius. This is due to thefact that the local perturbed pressure gradient is sufficient to self-sustain anexisting magnetic island. The main questions for burning plasmas are whetherthere is a trigger mechanism which will destabilize NTMs, and what is thebest strategy to control/avoid the modes. The latter has to take into accountthe main aim which is to maximize the Q factor, but also the controllabilityof the scenario. Standardized and simplified equations are proposed to enableeasier prediction of NTM control in burning plasmas from present experimentalresults. The present expected requirements for NTM control with localizedelectron cyclotron current drive (ECCD) in ITER are discussed in detail. Otheraspects of the above questions are also discussed, in particular the role of partialstabilization of NTMs, the possibility to control NTMs at small size with littleECH power and the differences between controlling NTMs at the resonantsurface or controlling the main trigger source, for the standard scenario namelythe sawteeth. It is shown that there is no unique best strategy, but several toolsare needed to most efficiently reduce the impact of NTMs on burning plasmas.

(Some figures in this article are in colour only in the electronic version)

4 Author to whom any correspondence should be addressed.5 Present address: ITER Organization, St Paul lez Durance, France.

0741-3335/10/025002+17$30.00 © 2010 IOP Publishing Ltd Printed in the UK 1

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

1. Introduction

Neoclassical tearing modes (NTMs) have been observed in many tokamaks [1, 2] and inparticular in H-modes (high confinement modes) with monotonic safety factor (q) profiles.This is due to the low marginal beta above which NTMs are metastable [3]. In H-modes, thebootstrap fraction is increased and the perturbed bootstrap fraction due to the localized islandis typically sufficient to sustain the island. The bootstrap current density is perturbed becauseof the local flattening of the pressure profile within the island, which also leads to a degradationof the confinement. The latter can be well estimated by the belt model [4], yielding a relationbetween the island full width w and the degradation of the energy confinement time τe:

�τe

τe

= �τ

wsat

a, with �τ = 4

ρ3s

a3, (1)

where wsat is the saturated island width, ρs the radius of the resonant surface and a the plasmaminor radius. NTMs can easily have island widths of 10% of the minor radius. With sucha width and if the mode is localized near mid-radius, the confinement degradation is of theorder of 5%. However, if the mode is near ρs/a = 0.8, the degradation is about 20%. Thisis why 2/1 NTMs, at q = 2, have much stronger impact on the plasma performance andclearly need to be avoided. Higher m/n modes, where m, n are the poloidal and toroidal modenumbers, respectively, are located closer to the center and therefore have smaller impact on theglobal plasma performance. This is why one might allow such modes to exist, although even4/3 modes have been observed to lead up to 10% degradation in the JET tokamak [5]. Notethat the island size can increase significantly if the mode locks and therefore lead to largerconfinement degradation [6–8].

Since NTMs are driven by a deficit of current density within the island, the easiest wayto stabilize them is by adding localized current with electron cyclotron current drive (ECCD)in the island [9–18]. Recently, a cross-machine comparison of ECCD stabilized NTMs hasallowed the prediction of the typical behavior expected in ITER [19]. It is shown that a drivencurrent density of the order of the local bootstrap current density should be sufficient to fullystabilize the mode, jcd/jbs ∼ 1, depending on the assumed model for the stabilizing mechanismat a small island size as will be discussed below. Such a criterion is important for the design ofthe EC launcher in ITER [20–22]. The range of validity of such a criterion is also importantand it will be discussed in detail in section 4. The aiming accuracy and related issues, as wellas the question of the 2/1 mode locking, have been analyzed in [7] and will not be analyzed indetails here, except with regard to the global strategy for NTM mitigation.

Another important aspect is the question of the existence of NTMs in burning plasmas andthe onset criteria. It has been demonstrated in JET that the main trigger mechanism in low betaplasmas is the sawtooth crash [23]. It is also shown that crashes after a long sawtooth periodeasily trigger NTMs, even at a very low beta near the marginal beta limit. Since fast particlesare very efficient to stabilize sawteeth [24, 25], it is assumed here that the main condition forthe existence of NTMs in sawtoothing burning plasmas will be related to the sawtooth activity.Therefore, the control of sawteeth is inherently part of any NTM control strategy. This will bediscussed in section 6. Note that for the other scenarios, hybrid and advanced, NTMs typicallyoccur at much higher βN values, close to the ideal limit, and both βN and q profiles influencethe onset conditions.

Other aspects related to NTMs and plasma performance are important. First theoptimisation should be related to the factor Q, the ratio between the fusion and the auxiliarypower. This factor characterizes the overall performance of a burning plasma. Since thestabilization of NTMs with ECCD tends to decrease Q, by increasing the auxiliary power, anoptimum Q might be obtained with partial stabilization of the NTM. Another parameter is the

2

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

relation between the time required to fully stabilize an NTM and the sawtooth period (or ratherthe time between successive NTM triggers). Finally, the possibility to use pre-emptive ECCD(ECCD applied before the NTM onset) to optimize the power required for NTM stabilizationwill also be discussed.

The paper is organized as follows, in section 2 we present the main operational diagramrelevant for NTM control in burning plasmas, namely the dependence of Q on additional ECpower. In section 3 the main equations for comparison with experiments and for the basisfor predictions are presented, in section 4 the criteria ηNTM = jcd/jbs for full stabilization areanalyzed, in section 5 predictions for Q and partial stabilization are presented and in section 6the strategy with respect to sawteeth activity and pre-emptive ECCD is discussed. Section 7concludes this paper.

2. Burning plasma conditions

In order to discuss the best strategies for NTM control, we use the effects of NTMs and/or ofthe auxiliary power required to stabilize them on the factor Q as a global measure of scenarioperformance. We use a simplified model to determine the burning temperature and totalpressure for a given auxiliary power and island width. We start from the so-called scenario2 of ITER [1], which is the baseline scenario for reaching Q = 10 in a sawtoothing ELMyH-mode. The main parameters of interest are R0 = 6.2 m, a = 2 m, Ip = 15 MA, B0 = 5.3 T,V = 830 m3, τE0 = 3.7 s, Zeff = 1.7, PNBI = 40 MW, Pα = 80 MW and PBrem = 21 MW.The scaling law assumed in this case yields τE ∼ P

−ePL with eP = 0.69 and PL the loss

power [1]. The fusion power is given by Pf = 5Pα with

Pα = γα1.5 × 10−6p2keVR (TkeV)V (MW), (2)

R(TkeV) = 29.84T 2.5keV exp

[− (TkeV + 0.11)0.45

0.43

], (3)

using a useful fit for the reactivity R and where pkeV = 1.6n19TkeV is the total pressure withn19 expressed in [1019 m−3] and T in [keV]. The total thermal energy is given by

WE = γE3.84 × 10−3 fpe n19TkeV V (MJ) = PLτE, (4)

with fpe = p/pe and where τE can be written as follows using the baseline parameters:

τE = τE0

(PL0

PL

)eP

(1 − �τw/a), (5)

with w the full island width and �τ given in equation (1). For the effective total heating power,we take into acount Bremsstrahlung radiation and the fact that any off-axis additional power islocated in a bad confinement region. Therefore, we weight its contribution by a profile effect.Assuming steady state we obtain

PL = Pα + PNBI +

(1 − ρ2

s

a2

)Pec − PBrem. (6)

In our case, given the position of the 2/1 surface, since the 2/1 mode is the main mode degradingplasma performance, we assume 1 − ρ2

s /a2 ≈ 0.5. The radiation term is important to limitthe benefits at large temperatures and is taken as follows:

PBrem = γB47.4 × 10−6Zeff n219V

√TkeV. (7)

3

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

The parameters γα , γE , γB are introduced to take into account the profile effects (since theplasma parameters in the above equations are taken at the plasma center). Assuming a flatdensity profile and T (ρ) ∼ (1 − ρ2) one gets

γα = 0.19, γE = 0.5, γB = 0.67. (8)

From equation (4) one obtains:

PL =[γE2.4 10−3fpe n19 TkeV V

τE0 PePL0(1 − �τ w/a)

] 11−eP

, (9)

which highlights the sensitivity on the power exponent in the scaling law. The burningtemperature is then obtained from equations (3), (6), (7) and (9). The measure of the globalburning plasma performance, Q = Pf/(PNBI + Pec), is then obtained from equation (3) at thistemperature. We have adjusted the factor fpe = p/pe, total pressure to electron pressure, to1.83 so as to essentially recover the standard steady-state conditions with no modes (w = 0):

Pα = 80 MW, Q = 10, Tburn ≈ 20 keV, βN ≈ 1.8,

PBrem ≈ 20 MW, (10)

with PNBI = 40 MW, PL0 = 99 MW, τE0 = HH 3.7 s and HH = 1. The HH factor representsany confinement improvement or degradation. With an additional 20 MW of EC power, weobtain Pα = 84 MW and Q = 7. In this way we can determine Pα for different HH valuesand additional Pec, still assuming no NTMs. The results are shown in figure 1 for HH valuesequally spaced between 0.75 and 1.25. This is the operational diagram of interest for thisstudy. We also show the ‘anchor’ points related to the 3/2 and 2/1 NTMs. If no ECCD isapplied, we expect a degradation of 15% (equation (1)), thus an effective HH = 0.85 for the3/2 mode. Similarly we predict HH = 0.75 for the 2/1 mode assuming it does not lock. If itwould lock, as predicted in [7] for w > 5–10 cm, the mode would grow to even larger valuesand the discharge would probably need to be stopped, although recent results show that thelocked mode can be controlled as well [26]. This yields Q = 6.9 with a stationary 3/2 modeand Q = 4.7 with a 2/1 mode (points A and B in figure 1). If the modes are fully stabilizedwith 20 MW and assuming that the EC power needs to be sustained in steady state, we recoverHH = 1 but at Pec = 20 MW, thus Q ≈ 7. Of course, if the EC power can be switched off,Q = 10 is recovered until the next appearance of an NTM if any. Increasing the EC power tostabilize the mode, one will move from points A or B in figure 1 to point D (or C if 10 MW issufficient to fully stabilize the mode), with a path to be determined and with possible valuessketched by the dashed lines. Since the dependence of the saturated island width wsat on Pec

can be relatively complex, one might find an optimum Q value at lower values of Pec andthus with a partially stabilized mode. These paths depend on the stabilizing parameters andpredicted marginal island widths. They will be calculated in section 5, but before that, oneneeds to define the model for calculating w(Pec) and to validate it with present experimentalresults. Depending on effective alignment and assumptions, the mode could be stabilized with5 MW of ECCD power [7]. In such a case, CW would be marginally acceptable since it wouldyield Q = 8.9.

3. Standard equations for comparison with experiments

The main characteristics of NTMs which have been observed experimentally are theproportionality of the saturated island width with poloidal beta, the self-stabilization at asmall island width, the confinement degradation (equation (1)), and the efficient stabilizationwith local ECCD. Other aspects have been observed, but most of them can be included in these

4

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

0 5 10 15 20 25 30

4

6

8

10

12

Pec [MW] at q=2

Q

HH=0.75

HH=1.25

B

A

C

D

Figure 1. Q versus additional EC power for HH values between 0.75 and 1.25. Points A and Bmark the predicted steady-state performance with either a 3/2 or a 2/1 NTM, respectively. PointsC and D assume full stabilization with constant 10 MW or 20 MW, respectively, and thus are onthe curve HH = 1. The dashed lines are a sketch of possible stationary operating points with a 3/2or a 2/1 mode partially stabilized.

global effects. Another important point is the fact that the exact value of the classical tearingparameter related to the total current density profile, ρs�

′, is neither easily measured nor welldefined. These characteristics are well encapsulated by the modified Rutherford equation.Since ρs|�′| is de facto a free parameter, it is better to normalize the equation for the islandgrowth rate by this value and one obtains the following two basic equations [3, 27]:

τR

ρ2s |�′|

dw

dt= −1 +

(1 − �τw

a

)wsat∞(βp)w

w2 + w2marg

− �′cd, (11)

τR

ρ2s |�′|

dw

dt= −1 +

(1 − �τw

a

)wsat∞(βp)

w

(1 − w2

marg

3w2

)− �′

cd, (12)

with

�′cd = cj

(1 − �τw

a

)wsat∞wcd

jcd

jbsηaux(w/wcd). (13)

The left-hand side represents the normalized island growth rate, where τR is the resistive time.The first term on the right-hand side is the stabilizing contribution from the equilibrium currentdensity at large island size. The latter is always negative at large island and should dependon w [28]. Here we assume ρs�

′ to be constant and negative, which is reasonable if the ratiobetween the largest saturated width and the marginal island width is not very large. However,a term like (1 − αw) should replace the term (−1) if the largest saturated width becomes verylarge [28].

The second term on the right-hand side is the driving term from the perturbed bootstrapcurrent: ρs�

′bs. It is proportional to 1/w at a large island size [29] and is reduced at a small

island size because the effective perturbed bootstrap current is not as large as if the pressurewas fully flattened. There are typically two forms which can explain the observed behaviorat small island size: equation (11) related to the effect of finite χ⊥/χ‖ [30], and equation (12)related to the effect of the polarization current [31]. The latter has been assumed in the recent

5

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

cross-machine analysis [19]. We propose to keep both options, since they have very differentbehavior at small w, the first one leads to ρs�

′bs ∼ w and the second to ρs�

′bs ∼ 1/w − 1/w3.

For full stabilization with ECCD, this is important because it takes place at a small islandsize. It is argued that using both forms proposed in equations (11) and (12) should spanthe effective dependence of the growth rate at small island width. The latter can indeed berather complicated once the effects on the curvature term [32] or of finite orbit width [33], forexample, are taken into account. Note that we use the simplest form which still takes the mainphysical effects. Therefore, for the second model, the frequency dependence is ignored andonly the main stabilizing contribution is taken into account with its w dependence. If the latterwould decrease due to frequency change, the first model would remain.

The values of wsat∞ and wmarg determine the strength of the driving term and of thestabilizing contributions at small island, respectively. The first relates to the saturated widthat a large island size. Indeed, for w � 1, both equations give the same saturated island width,without ECCD contribution:

wsat(dw/dt = 0, jcd = 0, wmarg ∼= 0) = wsat∞

1 + �τ

wsat∞a

, (14)

where the denominator includes self-consistently the reduction in βp due to the NTM andwsat∞ is the saturated island size without confinement degradation nor the stabilizing terms(wsat � wmarg, thus the subscript ‘∞’). The latter can be related to the usual terms of themodified Rutherford equation with

wsat∞ = ρsβpabs − aggj

ρs|�′| , (15)

where the bootstrap and ggj contributions are defined in [2] and the plasma parameters aretaken at the mode onset (or equivalently without the presence of a mode). Once the modeonsets, it is seen experimentally that the global βp decreases and the resultant saturated islandsize is smaller than that expected from the βponset value. Equations (14) and (15) give directlythe self-consistent saturated island width and the stationary βp = βponset/(1 + �τ

wsat∞a

). Forcross-machine comparison and prediction to ITER, relatively self-similar plasmas need tobe considered (here sawtoothing, ELMy H-modes for scenario 2) and the global βp is used.This has proven useful and robust in analyzing and predicting NTMs behavior in AUG [8],DIII-D [34], JET [3, 35, 36], MAST [37] and TCV [28] with the same method as definedin [2, 3]. Using ITER scenario 2 parameters, one obtains wsat∞ = 32 cm from equation (15)which leads to a predicted effective stationary saturated island width of 24 cm (equation (14))once the self-consistent confinement degradation is taken into account. This finally gives aneffective confinement degradation of 25%. For the 3/2 mode, we get wsat∞ = 25 cm andwsat = 21 cm from equation (14) for an effective confinement degradation of 15%. For the 2/1mode the expected saturated width without EC is larger than the width above which the modeis predicted to lock [7]. We shall discuss the implications in the last sections. The value hereis used to determine the characteristics near full stabilization.

The marginal island width wmarg is the island width at the maximum growth rate, withoutECCD contribution. Actually, at max(dw/dt) we have w � wmarg(1 − �τwmarg/a), but theconfinement degradation at w = wmarg is less than 5%. Thus, the main characteristics of themodified Rutherford equation are fully determined, without ECCD, once only two parametersare determined, namely wsat∞ and wmarg. This is shown in figure 2(a), where examples ofequations (11) and (12) are shown, with jcd = 0. Dashed–dotted lines show the island growthrate without the self-consistent confinement degradation (i.e. assumes �τ = 0) and thereforethe saturated island width is at w = wsat∞ (stable point with dw/dt = 0). On the other hand,including the �τ effect (solid lines) modifies the curves at large w and leads to the smaller

6

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

0 10 20 30 40–2

–1

0

1

2

3

4

5∆’

cd=0

(a)

wsat∞

wsat

wsat∞

=32

wsat∞

=wsat∞,marg

w [cm]

Normalized dw/dt

0 10 20 30 40–2

–1

0

1

2

3

4

5∆’

cd ≠ 0

w [cm]

Normalized dw/dt

(b)

Figure 2. RHS of equation (11) (blue online) or equation (12) (with a sharper drop at small w, redonline) with wmarg = 4 cm, wsat∞ = 32 cm and �τ = 2.05. (a) �′

cd = 0, with (solid) and without(dashed–dotted) the term �τ . The dashed lines are obtained with equation (17). (b) Same as solidlines of (a) including �′

cd assuming the 50% or CW with wcd = 2.5 cm (dashed) or 5 cm (solid)and 13.3 MW.

effective saturated island width wsat (equation (14)). This is due to the effect of the island onconfinement. If a mode onsets at βp = 0.66, it will degrade the confinement, βp will decreaseand the mode will not grow to the same size as if additional power would maintain βp = 0.66.This stationary effective wsat is directly obtained with the term �τ and fixed onset conditionsdetermined by wsat∞. The curves correspond to the predicted 2/1 mode in ITER Q = 10scenario with wsat∞ = 32cm, wsat = 24 cm, wmarg = 4 cm and thus �τwsat/a = 25%. It isalso interesting to note the value of wsat∞ required such that the maximum growth rate is zero,which determines the marginal beta limit [3]:

wsat∞,marg = 2wmarg for equation (11), (16)

wsat∞,marg = 1.5wmarg for equation (12). (17)

The dashed curves in figure 2(a) have been obtained with wsat∞ set to the above values. Inthis way, the ratio of wsat∞, measured with the onset βp, with 2wmarg or 1.5wmarg gives thehysteresis factor and is similar to the ratio βp,onset/βp,marg.

Experimentally, these two parameters, wsat∞ and wmarg, are easily measured. The firstone, wsat∞, is measured from the island width wsat when a sufficient βp is maintained. In thiscase, the effect of the confinement degradation is not negligible and �τ in equation (14) shouldbe taken into account, as seen from figure 2(a). The second parameter, wmarg, is obtained fromslow power ramp-down experiments [3, 8] from the island width when the mode quickly self-stabilizes. Note that a very slow decrease in βp is required to accurately measure wmarg [3].The ratio of wsat∞/wmarg should be of the order of 5 or larger, to ensure an accurate measureof the first parameter. Note that this is often not the case in ECCD experiments, where thesaturated island width before the ECCD is applied is often relatively small.

The last term on the right-hand side of equations (11) and (12) represents the contributiondue to localized ECCD, �′

cd = �′cd/|�′| (equation (13)). Here we assume perfect alignment,

while misalignment has been taken into account in [7, 19]. The coefficient cj is used to fit theexperimental results with the above equation. The driven current is assumed to be of the formj = jcd exp[−4(ρ − ρs)

2/w2cd], such that jcd is the peaked current density and wcd is the 1/e

full width. The function ηaux(w/wcd) is related to the stabilizing contribution of the parallel

7

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

current driven within the island and has typically the following dependence on w [13–15, 18]:

ηaux,cw

(x = w

wcd

)= 1

1 + 2x2/3, (18)

ηaux,50%

(x = w

wcd

)= 1.8

x2tanh

( x

2.5

). (19)

The fits are taken from [18], with a factor 1/2 for equation (19) to take into account the effectivetotal driven current and normalized such that ηaux ∼= 1 for w ∼= wcd. The first function,equation (18), assumes a localized CW deposition and the second, equation (19), a localizeddeposition near the O-point with 50% modulation. The effect of adding the last term on theright-hand side of equations (11) and (12) is shown in figure 2(b). An important characteristic isthat the w dependence of ηaux is not necessarily the same as the other contributions. Therefore,the maximum growth rate might shift to larger values than wmarg, which might have importanteffects when comparing various current density profile widths as will be discussed below. Thisis clearly seen in figure 2(b) where we have used the solid line case of figure 2(a) and added �′

cdwith jcd/jbs corresponding to the prediction for ITER with 13.3 MW and wcd = 2.5 or 5 cm.We see that we can essentially stabilize the mode and that the island width will self-stabilizeat about w = 10 cm (when max(dw/dt) = 0). This is indeed 2.5 times larger than wmarg andis a realistic situation.

Another form for ηaux,cw has been used in [27], ηaux,fs corresponding to equation (20)of [18], but this would not change much the results presented here. It should be noted thatin [18], when assuming a flux surface current density, the author neglected the fact that thephase α0 in equation (19) still depends on the flux surface label ψ . This will in turn define theeffective current density inside the island and gives values for ηaux in between ηaux,cw and ηaux,fs,depending on the assumed function α0(ψ). It is therefore better, for conservative assumptions,to take the form of equation (18) which was first calculated in [15]. The differences liewithin the error bar of present experimental results and the related predictions that one candraw.

4. Criteria for ηNTM ≡ jcd/jbs

Due to the high probability for ITER to have NTMs in its standard scenario [23] and theefficient stabilization obtained with ECCD [9–12], ITER will be equipped with EC upperlaunchers dedicated mainly to NTM stabilization. The design of such a launcher tries tomaximize the peaked jcd while minimizing the value of wcd. However, to evaluate whethera given design is predicted to be able to fully stabilize NTMs in ITER and moreover at whatpower level, it is useful to have a criterion for ηNTM ≡ jcd/jbs. At present, a criterion ofηNTM � 1.2 is used [21], based on [19, 20]. The latter is obtained assuming 50% modulationand the polarization model, equation (12). It does not take into account the depositionwidth, wcd, nor the marginal island width wmarg. Here, we analyze the effects of these twoparameters, in addition to assuming different models for the stabilizing terms at small islandwidth and CW versus 50% modulation. The aim is to obtain a more complete definition of thecriteria for ηNTM and a well-defined scheme to analyze the performance of a given launcherdesign.

4.1. Basic ITER parameters used for the study

Before analyzing the criteria for ηNTM, one has to determine the basic plasma parametersrequired in equations (11) and (12). That is, the expected values of wsat∞ and wmarg and the

8

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

Table 1. Parameters in equations (11) and (12) used to determine the values of cj required in bothequations, respectively, with wsat∞,χ from equation (11) and wsat∞,pol from equation (12).

Machine wmarg wsat∞,χ wsat∞,pol jcd/jbs wcd

AUG 1.5–1.8 3.7 2.9 3.1 1.1DIII-D 2.5 6.6 5.8 0.9 2.5JET 4 9.1 8.3 1.2 3.8JT-60U 4–5 12 10.5 1.2 11.2

value of cj to fit the present experiments. The value of �τ is given from the equilibriumused for the ray-tracing simulations. First the coefficient cj has been determined usingthe same method as in [20]. We have obtained the relevant parameters to reproduce theexperimental results presented in [19], using wcd = 6δec/5. They are given in table 1. Then,a coefficient cj is obtained such that all experiments in which full stabilization is realizedlead to max(dw/dt) < 0. Since there are large uncertainties in these early cross-machinecomparisons, we have taken ‘best estimates’ for the relevant coefficients. We have taken aconservative value of cj = 0.5 to be used with equation (11) and a value of cj = 1 forequation (12). Note that we have not considered the effects of local heating within the island,as has been recently observed in TEXTOR [38], nor the effect of jcd on �′ [39]. These effectsshould not be dominant in ITER, with good alignment, and they are in fact taken into account,to first order, with the coefficient cj . In addition to this coefficient, one should take into accountthe misalignment in the various experiments. However, the latter is not yet well determined,therefore the method to determine cj proposed here can be seen as purposedly conservativefor predictions to burning plasmas. We also assume that these coefficients, obtained from 3/2mode stabilization experiments, are also valid for 2/1 modes.

We now need to determine the plasma parameters expected in ITER. As mentioned earlier,we only need to predict two parameters, namely wsat∞ and wmarg. The latter has been predictedto be near 2 cm in [19] and 2–6 cm in [3]. We shall use three values, 2, 4 and 6 cm in orderto study its influence on the predicted results. On the other hand, the value of wsat∞ has notyet been studied systematically across machines, although it would also allow a good testof the theory by comparing with equation (15). Therefore, we have used the same methodas described in [2] and which has been used to compare with DIII-D [34], JET [3, 35, 36],MAST [37] and TCV [28]. In each case, the saturated island size was relatively well predictedusing global plasma parameters and the formulae in [2]. Using the parameters predicted for thestandard scenario in ITER, scenario 2 [1] with βp = 0.66 and βN = 1.8, we obtain for q = 2,a value of wsat∞ = 32 cm and ρs/a = 0.8 with a = 200 cm. The latter yields �τ = 2.05and an effective saturated island from equation (14) of wsat,2/1,ITER � 24 cm. Thus, one wouldhave an island of 12% the minor radius and a confinement degradation of 25% with βN = 1.8,which looks reasonable when compared with the present experimental results with rotating2/1 modes.

The latest design for the ITER upper launcher [22, 40] has a peaked current density with13.3 MW of jcd � 0.2 MA m−2, such that ηNTM � 2.7, and a width wcd � d/

√κ � 2.35 cm,

where d is the width defined with respect to the area [40]. The launcher is made of two rowsand the other row has a slightly larger value of wcd. This is why we have analyzed the resultsassuming wcd = 2.5, 5 and 10 cm in order to see the effects of larger deposition widths.Usually the total driven current is relatively constant, under similar launching conditions.Therefore, when considering various widths, one should keep (ηNTM wcd) constant. We shalluse ηNTM wcd = 6.3 to analyze the present launcher design, which corresponds to the valuepredicted with 13.3 MW (2/3 of the available power).

9

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

Table 2. Values of ηNTM expected to stabilize 2/1 NTMs on ITER. The first three rows correspondto different values of wmarg. The columns span the values of wcd = 2.5, 5 and 10 cm, as well as thetwo models considered using equation (11) for the columns labeled ‘χ⊥’ and using equation (12)for the columns labeled ‘pol’. The last two rows yield the average value per model and then percurrent density width wcd. Table for the CW case, using equation (18).

wcd = 2.5 cm wcd = 5 cm wcd = 10 cm

wmarg χ⊥ χpol χ⊥ χpol χ⊥ χpol

2 cm 3.7; 9.25 1.9; 4.75 2.5; 12.5 1.7; 8.5 4.4; 44 3.1; 314 cm 3.2; 8 1.8; 4.5 1.9; 9.5 1.1; 5.5 2.0; 20 1.5; 156 cm 2.6; 6.5 1.7; 4.25 1.5; 7.5 1.0; 5 1.3; 13 1.0; 10ηNTM; ηNTMwcd 3.17; 7.93 1.8; 4.5 1.97; 9.85 1.27; 6.35 2.57; 25.7 1.87; 18.7ηNTM; ηNTMwcd 2.49; 6.2 1.62; 8.1 2.22; 22.2ηNTMwcd 7.15

Table 3. Same as table 2 but assuming 50% modulation in the O-point, equation (19).

wcd = 2.5 cm wcd = 5 cm wcd = 10 cm

wmarg χ⊥ χpol χ⊥ χpol χ⊥ χpol

2 cm 3.0; 7.5 1.5; 3.75 2.1; 10.5 1.1; 5.5 2.0; 20 1.1; 114 cm 2.6; 6.5 1.5; 3.75 1.7; 8.5 1.0; 5.0 1.5; 15 0.9; 96 cm 2.1; 5.25 1.4; 3.5 1.3; 6.5 0.9; 4.5 1.1; 11 0.8; 8ηNTM; ηNTMwcd 2.57; 6.43 1.47; 3.68 1.7; 8.5 1.0; 5.0 1.53; 15.3 0.93; 9.3ηNTM; ηNTMwcd 2.02; 5.05 1.35; 6.75 1.23; 12.3ηNTMwcd 5.9

4.2. The criteria ηNTM

Using the above parameters into equation (11) or (12), with different values of wmarg (2, 4 or6 cm) and wcd (2.5, 5 or 10 cm), we can determine the value of ηNTM = jcd/jbs such that theNTM is unconditionally stable (max(dw/dt) < 0). These values are given in tables 2 and 3,with average values given in the three bottom rows. The value of the product ηNTMwcd isalso provided, since it is very useful when comparing various values of wcd. For the presentITER EC upper launcher design [22] with 13.3 MW, the expected value of ηNTM is 2.52, 1.26or 0.63 if wcd = 2.5, 5 or 10 cm, respectively. Looking at tables 2 and 3, we see that forwcd = 2.5 cm, in five cases out of six for 50% modulation and four out of six for CW, theNTM can be stabilized with the present design of the EC launcher. For wcd = 5 cm, 2/1 NTMscan be stabilized in four out of six test cases for both the CW option and the 50% modulation.Finally for wcd = 10 cm, NTMs cannot be stabilized in CW and marginally for 1 case using50% modulation (for wmarg = 6 cm). Assuming 20 MW, thus ηNTMwcd = 9.45, essentially allcases with wcd = 2.5 cm or 5 cm can be fully stabilized. From tables 2 and 3, we also see thatdepending on the model the 2/1 mode would be fully stabilized with EC power between 7 MWand 20 MW. Comparing wcd = 5 cm with 10 cm, we clearly see that localization is beneficialsince the required (ηNTMwcd) values decrease by a factor of two to three and the required ηNTM

is similar or smaller. On the other hand, when using 2.5 cm instead of 5 cm, the required valuesof ηNTM increase while the required ηNTMwcd are similar. This is due to the fact that wcd startsto be of the order of wmarg, so there is little gain by reducing it further. Actually, a high degreeof radial accuracy is also required for the ECCD deposition location, of the order of wcd orless [7]. Therefore, a value of wcd ≈ 5 cm could actually be an optimum.

10

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

Figure 3. Q factor obtained with a 2/1 mode controlled with PEC for various models, assumingwmarg = 2 cm, wsat∞ = 32 cm. The required values for full stabilization are given in tables 2and 3. (a) Assuming HH = 1, (b) with HH = 1.2.

Tables 2 and 3 can be used to compare between the stabilizing models. The requiredvalues for the ‘χpol’ model (equation (12)) are typically 50–70% of the ones for the ‘χ⊥’ model(equation (11)). However, a large part is due to the lower value of cj used in equation (11)which was very conservative. Using the same value as in equation (12) would lead to similarpredicted values in both cases. However, one needs further detailed experiments to betterquantify this value. Finally, we can compare the required values assuming CW (table 2) and50% modulation (table 3). We see that there are actually no significant differences except forwmarg = 2, since in this case w/wcd can be smaller than one, which is the domain where 50%modulation can be beneficial [12].

The criteria ηNTM � 1.2 was essentially based on the ‘pol’ model, equation (12), andassuming 50% modulation. With this model, we obtain a similar value, ηNTM � 1 fromtable 3 for the cases with wcd = 5 cm and independently of the value of wmarg. However ifwcd = 2.5 cm, then ηNTM � 1.5 should be used. Therefore, from tables 2 and 3 we see that amore general criteria should be

wcd � 5 cm and ηNTMwcd � 5 cm. (20)

5. Q predictions in the presence of finite widths NTMs and EC stabilization

The parameters given in tables 2 and 3 define the requirements for full stabilization. Asdiscussed in section 2 and shown in figure 1, one should consider the effective paths betweenno stabilization and full stabilization in order to determine the best strategy. This is performedin this section where we find the effective Q value for a given EC power and current driven,and a resulting saturated island size. In other words we solve equation (11) or (12) todetermine the saturated island size which determines through equations (1) and (3)–(9) theresulting Q factor. In this section we assume that the EC power is turned on all along. Thecase of stabilizing the mode and then switching off the EC power is considered in the nextsection.

We show in figure 3 the effective paths of a 2/1 NTM in ITER in the operational diagramQ versus EC power. We show the eight cases with wmarg = 2 cm and wcd = 2.5 or 5 cmcorresponding to the top row of tables 2 and 3. The order in the legend corresponds to

11

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

Figure 4. (a) Effective saturated island width with a given PEC as obtained from equations (11)and (12) corresponding to figure 3. (b) RHS of equation (12) for the case shown with a dashed linein (a) and figure 3.

increasing required (ηNTMwcd) values obtained in tables 2 and 3 for full stabilization. Forexample, for wcd = 2.5 cm, CW and assuming the χ⊥ model, table 2 gives ηNTMwcd = 9.25.Since the UL provide ηNTMwcd = 6.3 for 13.3 MW, this means that 19.5 MW are requiredfor full stabilization as confirmed by the sixth trace in figure 3(a). Figure 3(b) shows thesimilar situation but assuming an improved confinement factor HH = 1.2. It should be notedthat this only changes the Q values. The former example still stabilizes at 19.5 MW, butthis time at a Q value of 8.8 instead of 7. The latter effect can be inferred from figure 1as well.

Most cases shown in figure 3(a) have a slight Q increase with increasing PEC, until nearfull stabilization where Q(PEC) increases rapidly. With HH = 1.2, many cases have actuallyQ decreasing first until near full stabilization. In such cases the best is either full stabilizationor even to keep the mode as such with no EC power, in particular for HH > 1. Thesebehaviors can be understood from figure 4(a) which shows the predicted wsat versus PEC forthese eight cases. We see that the island width does not change much at first with small valuesof PEC. Therefore, there is little gain in confinement and a possible loss due to the additionalauxiliary power injected. However, when wsat reduces to near 15cm, there is a rapid variationwith increasing ECCD. This is because the effective wmarg with the �′

cd term is around 10 cmalthough we have wmarg = 2 cm in the model.

One case has a different behavior, the dashed line in figure 3, which is the case withwcd = 5 cm, CW and equation (12) (polarization model). In this case, there is a localmaximum of Q(PEC) at PEC < PEC,fullstab, that is at partial stabilization. This is of coursedue to a different dependence of wsat on PEC as seen from figure 4(a), the dashed line. In thiscase, wsat decreases rapidly at an intermediate power. To better understand the origin of thisbehavior, we plot the right-hand side of equation (12) with increasing PEC from 0 to 30 MW(figure 4(b)). With increasing �′

cd, the growth rate has a relatively flat dependence on w forw > 5 cm. This is why for a small power increase, for example between 10 and 14 MW, wsat

shrinks from 15 to 5 cm (solid circles), yielding a confinement degradation �τwsat/a of 15%down to 5% only. On the other hand, since wmarg is small, the peak of the normalized growthrate is still relatively high and one needs an extra 4 MW to shrink wsat from 5 to 2.5 cm whereit self-stabilizes. These effects could happen more often than expected once all the terms andtheir various w dependences are taken into account in the modified Rutherford equation. It

12

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

Figure 5. (a) Time evolution predicted for a 2/1 mode in ITER, assuming a 7 cm and a 24 cm seedisland, using equations (11) and (12) with 13.3 MW (ηNTMwcd = 6.3) and the models indicated inthe legend in order of the required jcd for full stabilization. (b) Using the 1st, 4th and 5th modelsof (a), the time evolution of a 7 cm seed island with 0 s, 3 s, 10 s and 20 s delay for the CD term tobe included.

also shows that the dependence of wsat on PEC is neither linear nor smooth, and this couldcomplicate some experimental observations.

6. Sawteeth and pre-emptive stabilization

In the previous section, we have seen that it is not always the best option, in terms of globalinstantaneous performance, to fully stabilize an NTM. In this section we study whether it couldbe beneficial to turn on the EC power only part of the time, in order to reduce the EC energyinjected into the plasma and therefore improve the integrated performance. This dependsmainly on the frequency of the trigger mechanism, the size of the seed island and on the timeit takes to fully stabilize the mode. It could be that the mode is triggered only in the initialphase of the scenario development, due to a first long sawtooth period or a large first ELM asit is seen in JET. In that case, the best strategy is clearly to fully stabilize the mode, with all thepower available and then to turn off the EC power while recovering Q = 10. Since sawteethwill happen regularly in the stationary ELMy H-mode scenario 2 case, a regular trigger mightoccur at every sawtooth crash. The sawtooth period can be in between 10 s and 100 s in ITER,but is hard to predict, and this will influence the best strategy. On the other hand, the time ittakes to fully stabilize the mode can be predicted using equation (11) or (12) with equation (1).This is shown in figure 5(a) assuming two seed island sizes of 7 cm and 24 cm. The first sixcases of figure 3 are shown, in order of the required (ηNTMwcd) values (tables 2 and 3), thistime for the cases with wmarg = 4 cm. We see that all cases shown can rapidly, within 6 s,fully stabilize the mode with 13.3 MW if the seed island is about 7 cm. However, in two cases(which have (ηNTMwcd) > 6.3 in tables 2 and 3) the mode cannot be stabilized if the seedisland is 24 cm. Note that it still takes between 27 s and 66 s to fully stabilize the mode. Thiscan be seen as a reason for the advantage of pre-emptive ECCD (which consists of having ECturned on before the mode is triggered) which will catch the mode in its early stage if it growsfrom a small island size. Note that the seed island could be created directly at a very large size

13

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

at the sawtooth crash during the reconnection process as observed sometimes in JET [3]. Inthat case, pre-emptive ECCD would only be beneficial if it modifies the formation of the seedisland during the sawtooth crash.

Another aspect to be taken into account to choose the best strategy on ITER is the fact thatthe resistive time is very long as compared with the present tokamaks. Therefore the NTMgrowth is very slow, so one might catch the mode early even if the EC power is turned on onlyafter the mode is detected. This is tested in figure 5(b) where a seed island of 7 cm is assumedto be triggered at t = 0 and then the EC power is turned on at t = 0 s, 3 s, 10 s or 20 s. The0 s delay would correspond to pre-emptive ECCD, while the other cases to the time it takesto detect the mode, aim at q = 2 and turn on the power. A 3 s delay time is a reasonablevalue if the launchers are already aiming at the q = 2 surface (from real-time equilibriumand ray-tracing calculations). In this case, three models are displayed, corresponding to thefirst, fourth and fifth cases shown in figure 5(a) (with corresponding colors). With 0 s delay,all models can fully stabilize the mode, as seen above. However already with a 3 s delay, onemodel cannot stabilize the mode because of the insufficient value of (ηNTMwcd). If one canfully stabilize the mode, with the other two models shown, then it takes 8–26 s if the EC isturned on after 3 s, 23–55 s if there is a delay of 10 s, and 39–75 s if there is a delay of 20 s.These simulations show that there is a large increase in efficiency if the island is caught when itis below about 10–15 cm. This relates to the effective marginal island size observed in section 5of about 10 cm when �′

cd is included even if wmarg = 2 cm in the model. Another result is thatit can easily take more than 20 s to fully stabilize the mode. Therefore, if the sawtooth periodis below about 30 s, it is clearly better to keep EC on all the time at either full or partial powerdepending on the effective Q value obtained. On the other hand, if the sawtooth period is verylong, then it might be better to turn on the EC power for a short time of about 10–20 s. Actually,real-time measurements and calculations should allow us to predict the next sawtooth crashand allow the EC to be turned on just before it occurs, yielding the 0 s delay case of figure 5(b).This could become the best strategy if the sawtooth period is longer than 30–60 s. This iseven more true if the mode locks when w < 5–10 cm as predicted in [7]. In that case, it isvery important to catch the mode before it grows to that size, although promising stabilizationof a locked mode has been recently obtained [26]. From figure 5(b), we see that this meansthe EC power should be active in less than about 3 s, which means effectively pre-emptiveECCD.

7. Conclusions

The plasma performance of a burning plasma, characterized by the factor Q = Pfusion/Paux, isanalyzed in detail with respect to the presence of NTMs. A finite island width leads to a locallyenhanced radial transport and to a finite confinement degradation. This effect is proportionalto ρ3

s , where ρs is the radius of the q = m/n surface (equations (1)). Therefore the 2/1 mode,with ρs ≈ 0.8, is the most critical and we have concentrated our study on this mode.

The NTMs can effectively be stabilized by driving local current density within the island[9–12]. This shrinks the saturated island and the plasma recovers better confinement. On theother hand, the additional auxiliary power decreases Q (figure 1), except if Pfusion increasessignificantly. The effective dependence of Q on PEC for the 2/1 mode on ITER, assumingexact aiming and no mode locking, has been calculated (section 5). It is shown that in mostcases there is no much gain, or even a decrease in Q, at low PEC, up to near full stabilization.On the other hand, in some cases partial stabilization can lead to better Q factor. This is dueto a rapid reduction in the island width with small input power, while the peak of the islandgrowth rate, at a small island width, still needs significant power to be decreased below zero.

14

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

This ‘non-linear’ dependence of wsat on PEC could be observed in the present experimentswhere more stabilizing terms are at play with similar amplitude and different w dependence.This effect could play a role and complicate comparison of experimental results acrossmachines.

A key element in these discussions is of course the power required for full stabilization.We have analyzed the 2/1 mode predicted in scenario 2 of ITER with respect to two modelequations, three different current drive widths, wcd, and three different expected marginal islandwidths. These model equations, equations (11) and (12), are proposed to be used to comparewith the present experimental results. They have been written such that only three mainparameters (wsat∞, wmarg and cj ) need to be determined and compared with the experimentalsituation and such that it is most appropriate for analyzing stationary conditions as obtainedexperimentally. They represent adequately the driving term (with wsat∞), the plasma stabilizingterm (wmarg) and the effect of EC (with cj ). Note that even though heating effects, CD effectson �′ and other terms are not explicitly included, they are taken into account implicitly throughthese three terms. This is in particular useful for cross-machine comparisons. We have seenthat the comparison of the �′

cd term with experimental results depends on the model equationused, leading to possibly different matching parameters cj in equation (13).

The required values of jcd for full stabilization of the 2/1 mode in ITER have beencalculated. They are obtained from comparing recent experimental results [19] withequations (11) and (12) and inferring the main parameters wsat∞ and wmarg for each machine(table 1). The minimum values of ηNTM = jcd/jbs and (ηNTMwcd) are provided in tables 2and 3. This allows us to better check whether a given launcher design is able to fully stabilizethe 2/1 mode in ITER. For example, the present design yields ηNTM ≈ 2.5 for a typical wcd of2.5 cm, giving a value of ηNTMwcd = 6.3 with 13.3 MW. The tables show that this is sufficientto fully stabilize the mode, even with wcd = 5 cm to take into account some misalignments ofthe various beams, especially if 20 MW are used. It also shows that a criterion like ηNTM � 1.2is very simple [21] and both ηNTM and (ηNTMwcd) need to be considered as well as the variousmodels. The results presented in tables 2 and 3 lead to the criterion (equation (20)): wcd � 5 cmand ηNTMwcd � 5 cm.

Figure 1 shows that full stabilization with continuous 13.3 or 20 MW leads to a Q ofabout 7.7 or 7, respectively. Therefore, the best strategy cannot be to keep the full powerrequired to stabilize the mode turned on continuously. The analysis of the time it takes tofully stabilize the mode depending on the seed island width and depending on the delay beforethe ECCD is turned on at the island location helps to determine the best strategies for NTMcontrol. We have shown that if the mode is captured before it reaches about 10 cm, then it ismuch easier to stabilize the mode and much faster. It can be performed in 3–10s, dependingon the model. However with a 3–10 s delay or if the mode has reached about 20 cm, then itcan easily take up to 60 s to get rid of the mode. Note that it takes 3 s for a mode triggered ata seed island size of 7 cm to reach 12 cm (figure 5). This is why pre-emptive ECCD coupledwith real-time calculations to determine the time of the next sawtooth crash might be the beststrategy to minimize the integrated EC power necessary for NTM control. On the other hand,this will depend on the sawtooth period itself and part of the power might be used to controlthe sawtooth period. It has been shown in [41] that the sawtooth period can be reduced byabout 30% with ECCD well inside the q = 1 surface, while it can be increased by 50% with areal-time control algorithm, similar to NTM control, such that the current is driven just outsidethe q = 1 surface with the upper launcher. Thus a combination of sawtooth control and NTMpre-emptive control might be the best. Note that the role of pre-emptive ECCD on the effectivesize of the seed island triggered at a sawtooth crash after a long sawtooth period has not beenstudied experimentally yet. This is important to know in order to predict the seed island sizes

15

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

expected on ITER. The goal of stabilizing/controlling the mode before it reaches 10 cm is alsoin line with the critical width above which the mode is expected to lock [7]. If it locks, theexcursions in confinement (and β, li) will be probably very large in a burning plasma and amachine like ITER or DEMO and should be avoided. Therefore, pre-emptive ECCD withreal-time control to determine when EC will be necessary appear again as the best strategyin order to maximize Q. On the other hand, for the hybrid and advanced scenarios, the mainsource of NTMs is due to the q and pressure profiles evolution. Therefore in these cases, theEC power might need to be turned on continuously to avoid NTMs and the discussion relatedto figure 1 describes the operational diagram for these scenarios.

The studies presented here show that the requirements to control NTMs in burning plas-mas are not just a question of sufficient local current drive capabilities and of aiming at theright position. The role of PEC on Q, the frequency of the trigger events, the size of the seedisland, the time it takes to fully stabilize the mode and even the dependence of the drivingand stabilizing terms on the island width play a role in determining the best strategy for NTMcontrol.

©Euratom 2010.

References

[1] Progress in the ITER Physics Basis 2007 Nucl. Fusion 47 S1[2] Sauter O et al 1997 Phys. Plasmas 4 1654 and references therein[3] Sauter O et al 2002 Plasma Phys. Control. Fusion 44 1999[4] Chang Z and Callen J D 1990 Nucl. Fusion 2 219[5] Buttery R J et al 2004 Proc. 20th IAEA Conf. on Fusion Energy Conf. (Vilamoura, Portugal, 2004) (Vienna:

IAEA) paper IAEA-CN-116/EX/7-1[6] Hender T C et al 2004 Nucl. Fusion 44 778[7] La Haye R et al 2009 Nucl. Fusion 49 045005[8] Maraschek M et al 2003 Plasma Phys. Control. Fusion 45 1369[9] Gantenbein G et al 2000 Phys. Rev. Lett. 85 1242

[10] La Haye R J et al 2002 Phys. Plasmas 9 2051[11] Isayama A et al 2007 Nucl. Fusion 47 773[12] Maraschek M et al 2007 Phys. Rev. Lett. 98 025005[13] Hegna C C and Callen J D 1997 Phys. Plasmas 4 2940[14] Zohm H 1997 Phys. Plasmas 4 4333[15] Perkins F W et al 1997 Proc. 24th European Conf. on Controlled Fusion and Plasma Physics (Berchtesgaden,

Germany, 1997) (Petit Lancy: European Physical Society) vol 21A (ECA) p 1017[16] Giruzzi G et al 1999 Nucl. Fusion 39 107[17] Ramponi G et al 1999 Phys. Plasmas 6 3561[18] Sauter O 2004 Phys. Plasmas 11 4808[19] La Haye R et al 2006 Nucl. Fusion 46 451[20] Zohm H et al 2006 14th Workshop on ECE and ECRH (Santorini, Greece)[21] Saibene G et al 2006 Proc. 21st IAEA Conf. on Fusion Energy Conf. (Chengdu, China, 2006) (Vienna: IAEA)

paper IT/P2-14[22] Henderson M A et al 2006 Proc. 21st IAEA Conf. on Fusion Energy Conf. (Chengdu, China, 2006) (Vienna:

IAEA) paper IT/P2-15[23] Sauter O et al 2002 Phys. Rev. Lett. 88 105001[24] Porcelli F et al 1996 Plasma Phys. Control. Fusion 38 2163[25] Eriksson L-G et al 2004 Phys. Rev. Lett. 92 235004[26] Volpe F A G et al 2009 Phys. Plasmas 16 102502[27] Sauter O et al 2006 Proc. 21st IAEA Conf. on Fusion Energy Conf. (Chengdu, China, 2006) (Vienna: IAEA)

paper TH/P3-10[28] Reimerdes H et al 2002 Phys. Rev. Lett. 88 105005[29] Carrera R et al 1986 Phys. Fluids 29 899

16

Plasma Phys. Control. Fusion 52 (2010) 025002 O Sauter et al

[30] Fitzpatrick R 1995 Phys. Plasmas 2 825[31] Mikhailovskii A B 2003 Contrib. Plasma Phys. 43 125 and references herein[32] Lutjens H, Luciani J-F and Garbet X 2001 Phys. Plasmas 8 4267[33] Poli E et al 2002 Phys. Rev. Lett. 88 075001[34] La Haye R J and Sauter O 1998 Nucl. Fusion 38 987[35] Huysmans G T A et al 1999 Nucl. Fusion 39 1965[36] Buttery R J et al 2000 Plasma Phys. Control. Fusion 42 B61[37] Buttery R et al 2002 Phys. Rev. Lett. 88[38] Westerhof E et al 2007 Nucl. Fusion 47 85[39] La Haye R J et al 2008 Nucl. Fusion 48 054004[40] Ramponi G et al 2008 Nucl. Fusion 48 054012[41] Zucca C 2009 Modeling and control of the current density profile in tokamaks and its relation to electron transport

PhD Thesis No. 4360, EPFL, Lausanne http://library.epfl.ch/theses/?nr=4360

17